Derivative of Constant

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The derivative of constant function is zero. That is , if 'c' is a real number, then

$ \frac{\text{d}[c]}{\text{d}x} = 0$

This shows that the slope is zero that means the line is parallel to the X-axis.
So the equation of the line is y= b( b is any y co-ordinate.
slope of horizontal line


slope = m =$\frac{y2-y1}{x2 -x1}$

slope of AB =$\frac{3-3}{5 -1}$

slope of AB =$\frac{0}{4}$

slope of AB =0
As the slope is zero so the equation of the line AB is y = 3( as the line passes through y=3)

slope = m =$\frac{y2-y1}{x2 -x1}$

slope of AC =$\frac{3-3}{1 -(-1)}$

slope of AC =$\frac{0}{2}$

slope of AC =0
As the slope is zero so the equation of the line AC is y = 3( as the line passes through y=3)

Prove that the derivative of a constant function is zero using the limit definition.
Proof : Let f(x) = y = c ( 'c' be any real number)
f(x + $\triangle x $) = c ( as there is no x in the function)
According to the definition of the derivative of the function

$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}\frac{f(x + \triangle x) - f(x)}{\triangle x}$

Plug in all the values

$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}\frac{c - c}{\triangle x}$

$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}\frac{0}{\triangle x}$

$\frac{\text{d}y}{\text{d}x}=\lim_{\triangle x \rightarrow 0}0$

= 0

Examples of derivative of constant

Function : 

1) f(x) = 4

2) f(x) = -3

3) s(t) = π

4) y = 0.8

5) y= -3/4

Derivative:

1) f'(x) = 0

2) f'(x) = 0

3) s'(t) =0 (since π is constant)

4) y' = 0

5) y' = 0


12th grade math

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